New bounds for numbers of primes in element orders of finite groups

Let $\rho (n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $\rho (n)\le 5n$ for all $n\ge 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.